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Hydromechanics of low-Reynolds-number flow. Part 3. Motion of a spheroidal particle in quadratic flows

Published online by Cambridge University Press:  29 March 2006

Allen T. Chwang
Allen T. Chwang
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge Permanent address: Engineering Science Department, California Institute of Technology, Pasadena.
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Abstract

Exact solutions in closed form have been found using the singularity method for various quadratic flows of an unbounded incompressible viscous fluid at low Reynolds numbers past a prolate spheroid with an arbitrary orientation with respect to the fluid. The quadratic flows considered here include unidirectional paraboloidal flows, with either an elliptic or a hyperbolic velocity distribution, and stagnation-like quadratic flows as typical representations. The motion of a force-free spheroidal particle in a paraboloidal flow has been determined. It is shown that the spheroid rotates about three principal axes with angular velocities governed by a set of Jeffery orbital equations with the rate of shear evaluated at the centre of the spheroid. These angular velocities depend on the minor-to-major axis ratio of the spheroid and its instantaneous orientation, but are independent of its actual size. The spheroid also translates at a variable speed, depending on its orientation relative to the surrounding fluid, along a straight path parallel to the main flow direction without any side drift or migration. This ‘jerk’ motion obeys a trajectory equation which is size dependent.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 72 , Issue 1 , 11 November 1975 , pp. 17 - 34
Copyright
© 1975 Cambridge University Press

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References

Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Chwang, A. T. & Wu, T. Y. 1974 Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies. J. Fluid Mech. 63, 607–622.Google Scholar
Chwang, A. T. & Wu, T. Y. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67, 787–815.Google Scholar
Edwardes, D. 1892 Steady motion of a viscous liquid in which an ellipsoid is constrained to rotate about a principal axis. Quart. J. Math. 26, 70–78.Google Scholar
Hancock, G. J. 1953 The self-propulsion of microscopic organisms through liquids. Proc. Roy. Soc. A 217, 96–121.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. A 102, 161–179.Google Scholar
Oberbeck, A. 1876 Ueber stationäre Flüissigkeitsbewegungen mit Berücksichtigung der inneren Reibung. J. reine angew. Math. 81, 62–80.Google Scholar
Simha, R. 1936 Untersuchungen über die Viskosität von Suspensionen und Lösungen. Kolloid Z. 76, 16–19.Google Scholar
Wu, T. Y. & Chwang, A. T. 1974 Double-model flow theory - a new look at the classica problem. Proc. 10th ONR Symp. Naval Hydrodyn., Cambridge, Mass.